= Shared zero-inflated Gamma-Poisson count model
{title2=$Y_j\mid B\sim\operatorname{Pois}(B\mu_j)$}
Let $B=0$ with probability $\pi$, and otherwise let $B$ have a <Gamma distribution> with mean one and <variance> $\tau$. Given $B$, counts $Y_j$ are <conditionally independent> <Poisson random variables> with means $B\mu_j$. For $q=1-\pi$, the <law of total variance> and <law of total covariance> give
$$
\mathbb EY_j=q\mu_j,\qquad\operatorname{Cov}(Y)=\operatorname{diag}(q\mu)+q(\tau+\pi)\mu\mu^T.
$$
Each component has a <zero-inflated negative binomial distribution>, but the components are not independent after marginalizing $B$. The zero component is shared by the whole profile. At $\tau=0$, interpret the positive Gamma component as a <point mass> at one.
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